Changes in the hyperelastic properties of endothelial cells induced by tumor necrosis factor-alpha

Mechanical properties of living cells can be determined using atomic force microscopy (AFM). In this study, a novel analysis was developed to determine the mechanical properties of adherent monolayers of pulmonary microvascular endothelial cells (ECs) using AFM and finite element modeling, which considers both the finite thickness of ECs and their nonlinear elastic properties, as well as the large strain induced by AFM. Comparison of this model with the more traditional Hertzian model, which assumes linear elastic behavior, small strains, and infinite cell thickness, suggests that the new analysis can predict the mechanical response of ECs during AFM indentation better than Hertz's model, especially when using force-displacement data obtained from large indentations (>100 nm). The shear moduli and distensibility of ECs were greater when using small indentations (<100 nm) compared to large indentations (>100 nm). Tumor necrosis factor-α induced changes in the mechanical properties of ECs, which included a decrease in the average shear moduli that occurred in all regions of the ECs and an increase in distensibility in the central regions when measured using small indentations. These changes can be modeled as changes in a chain network structure within the ECs.     

Nano-mechanical Compliance of Müller Cells Investigated by Atomic Force Microscopy

It has been known that a single Müller cell displays a large variation in the cytoskeletal compositions along its cell body, suggesting different mechanical properties in different segments. Müller cells are thought to be involved in many retinal diseases such as retinoschisis, which can be facilitated by a mechanical stress. Thus, mapping of mechanical properties on localized nano-domains of Müller cells could provide essential information for understanding their structural functions in the retina and roles in their pathological progresses. Using Atomic Force Microscopy (AFM) - based bio-nano-mechanics, we have investigated the local variations of the mechanical properties of Müller cells in vitro. We have a particular interest in identifying elastic moduli in regions closer to three distinctive segments of the cells - process, endfoot, and soma. Using the modified spherical AFM probes, we were able to accurately determine mechanical properties, i.e., elastic moduli from the obtained force-distance curves. We found that the regions closer to soma were mechanically more compliant than regions closer to endfoot and process of Müller cells. We found that this lateral heterogeneity of the mechanical compliance within a single Müller cell is consistent with reports from other cell types. The local variation in mechanical compliances along a single Müller cell may support their diverse mechanical functions in the retina such as a soft mechanical embedding, mechanosensing, and neurotrophic functions for neurons.     

An application of the micropipette technique to the measurement of the mechanical properties of cultured bovine aortic endothelial cells

The mechanical properties of endothelial cells were measured using the micropipette technique. The cells employed were collected from bovine aortic endothelium and cultured in our laboratory. Endothelial cells from confluent monolayers under no-flow conditions were detached from their substrate by trypsin or by a mechanical method and suspended in modified Dulbecco medium (MDM). In the micropipette technique, a part of the cell is aspirated into the tip of the micropipette under a microscope, and the deformation measured from a photograph. In this study, the data obtained were analyzed using a model where the cytoskeletal elements, which are considered to be the primary stress bearing components, are assumed to reside in a submembranous, cortical layer. Detached cells were found to have almost homogeneous mechanical properties based on measurements from different regions of the surface of a single cell. However, a hysteresis loop was observed in the relation between pressure and cell deformation during the loading and unloading processes. The calculated elastic shear moduli obtained for the trypsin-detached cells were as much as 10-20 times larger than those of a red blood cell. Mechanically-detached cells had moduli approximately twice that of the trypsin detached cells. Passage time, i.e., cell culture age, had no influence on the mechanical properties of the trypsin-detached cells, but did have an effect on the mechanically-detached cells, with both the younger and older cells being somewhat stiffer.     

Viscoelastic properties of human mesenchymally-derived stem cells and primary osteoblasts, chondrocytes, and adipocytes

The mechanical properties of single cells play important roles in regulating cell-matrix interactions, potentially influencing the process of mechanotransduction. Recent studies also suggest that cellular mechanical properties may provide novel biological markers, or "biomarkers," of cell phenotype, reflecting specific changes that occur with disease, differentiation, or cellular transformation. Of particular interest in recent years has been the identification of such biomarkers that can be used to determine specific phenotypic characteristics of stem cells that separate them from primary, differentiated cells. The goal of this study was to determine the elastic and viscoelastic properties of three primary cell types of mesenchymal lineage (chondrocytes, osteoblasts, and adipocytes) and to test the hypothesis that primary differentiated cells exhibit distinct mechanical properties compared to adult stem cells (adipose-derived or bone marrow-derived mesenchymal stem cells). In an adherent, spread configuration, chondrocytes, osteoblasts, and adipocytes all exhibited significantly different mechanical properties, with osteoblasts being stiffer than chondrocytes and both being stiffer than adipocytes. Adipose-derived and mesenchymal stem cells exhibited similar properties to each other, but were mechanically distinct from primary cells, particularly when comparing a ratio of elastic to relaxed moduli. These findings will help more accurately model the cellular mechanical environment in mesenchymal tissues, which could assist in describing injury thresholds and disease progression or even determining the influence of mechanical loading for tissue engineering efforts. Furthermore, the identification of mechanical properties distinct to stem cells could result in more successful sorting procedures to enrich multipotent progenitor cell populations.     

Active Prestress Leads to an Apparent Stiffening of Cells through Geometrical Effects

Tuning of active prestress, e.g., through activity of molecular motors, constitutes a powerful cellular tool to adjust cellular stiffness through nonlinear material properties. Understanding this tool is an important prerequisite for our comprehension of cellular force response, cell shape dynamics, and tissue organization. Experimental data obtained from cell-mechanical measurements often show a simple linear dependence between mechanical prestress and measured differential elastic moduli. Although these experimental findings could point to stress-induced structural changes in the material, we propose a surprisingly simple alternative explanation in a theoretical study. We show how geometrical effects can give rise to increased cellular force response of cells in the presence of active prestress. The associated effective stress-stiffening is disconnected from actual stress-induced changes of the elastic modulus, and should therefore be regarded as an apparent stiffening of the material. We argue that new approaches in experimental design are necessary to separate this apparent stress-stiffening due to geometrical effects from actual nonlinearities of the elastic modulus in prestressed cellular material.     

Contribution of the nucleus to the mechanical properties of endothelial cells.

The cell nucleus plays a central role in the response of the endothelium to mechanical forces, possibly by deforming during cellular adaptation. The goal of this work was to precisely quantify the mechanical properties of the nucleus. Individual endothelial cells were subjected to compression between glass microplates. This technique allows measurement of the uniaxial force applied to the cell and the resulting deformation. Measurements were made on round and spread cells to rule out the influence of cell morphology on the nucleus mechanical properties. Tests were also carried out with nuclei isolated from cell cultures by a chemical treatment. The non-linear force-deformation curves indicate that round cells deform at lower forces than spread cells and nuclei. Finite-element models were also built with geometries adapted to actual morphometric measurements of round cells, spread cells and isolated nuclei. The nucleus and the cytoplasm were modeled as separate homogeneous hyperelastic materials. The models simulate the compression and yield the force-deformation curve for a given set of elastic moduli. These parameters are varied to obtain a best fit between the theoretical and experimental data. The elastic modulus of the cytoplasm is found to be on the order of 500N/m(2) for spread and round cells. The elastic modulus of the endothelial nucleus is on the order of 5000N/m(2) for nuclei in the cell and on the order of 8000N/m(2) for isolated nuclei. These results represent an unambiguous measurement of the nucleus mechanical properties and will be important in understanding how cells perceive mechanical forces and respond to them.     

Stiffness of MDCK II Cells Depends on Confluency and Cell Size

Mechanical phenotyping of adherent cells has become a serious tool in cell biology to understand how cells respond to their environment and eventually to identify disease patterns such as the malignancy of cancer cells. In the steady state, homeostasis is of pivotal importance, and cells strive to maintain their internal stresses even in challenging environments and in response to external chemical and mechanical stimuli. However, a major problem exists in determining mechanical properties because many techniques, such as atomic force microscopy, that assess these properties of adherent cells locally can only address a limited number of cells and provide elastic moduli that vary substantially from cell to cell. The origin of this spread in stiffness values is largely unknown and might limit the significance of measurements. Possible reasons for the disparity are variations in cell shape and size, as well as biological reasons such as the cell cycle or polarization state of the cell. Here, we show that stiffness of adherent epithelial cells rises with increasing projected apical cell area in a nonlinear fashion. This size stiffening not only occurs as a consequence of varying cell-seeding densities, it can also be observed within a small area of a particular cell culture. Experiments with single adherent cells attached to defined areas via microcontact printing show that size stiffening is limited to cells of a confluent monolayer. This leads to the conclusion that cells possibly regulate their size distribution through cortical stress, which is enhanced in larger cells and reduced in smaller cells.     

The structure and mechanical properties of the proteins of lamprey cartilage

The cyanogen bromide‐resistant proteins of lamprey cartilage are biochemically related to the mammalian elastic protein, elastin. This study investigates their mechanical properties and enquires whether, like elastin, long‐range elasticity arises in them from a combination of entropic and hydrophobic mechanisms. Branchial and pericardial proteins resembled elastin mechanically, with elastic moduli of 0.13–0.35 MPa, breaking strains of 50%, and low hysteresis. Annular and piston proteins had higher elastic moduli (0.27–0.75 MPa) and larger hysteresis. Exchanging solvent water for trifluoroethanol increased the elastic moduli, whereas increasing temperature lowered the elastic moduli. Raman microspectrometry showed small differences in side‐chain modes consistent with reported biochemical differences. Decomposition of the amide I band indicated that the secondary structures were like those of elastin, preponderantly unordered, which probably confer the conformational flexibility necessary for entropy elasticity. Piston and annular proteins showed the strongest interactions with water, suggesting, together with the mechanical testing data, a greater role of hydrophobic interactions in their mechanics. Two‐photon imaging of intrinsic fluorescence and dye injection experiments showed that annular and piston proteins formed closed‐cell honeycomb structures, whereas the branchial and pericardial proteins formed open‐cell structures, which may account for the differences in mechanical properties. © 2014 Wiley Periodicals, Inc. Biopolymers 103: 187–202, 2015.     

The role of prestress and architecture of the cytoskeleton and deformability of cytoskeletal filaments in mechanics of adherent cells: a quantitative analysis.

Mechanical properties of adherent cells were investigated using methods of engineering mechanics. The cytoskeleton (CSK) was modeled as a filamentous network and key mechanisms and corresponding molecular structures which determine cell elastic behavior were identified. Three models of the CSK were considered: open-cell foam networks, prestressed cable nets, and a tensegrity model as a special case of the latter. For each model, the modulus of elasticity (i.e. an index of resistance to small deformation) was given as a function of mechanical and geometrical properties of CSK filaments whose values were determined from the data in the literature. Quantitative predictions for the elastic modulus were compared with data obtained previously from mechanical tests on adherent cells. The open-cell foam model yielded the elastic modulus (10(3)-10(4)Pa) which was consistent with measurements which apply a large compressive stress to the cell. This suggests that bending of CSK filaments is the key mechanism for resisting large compression. The prestressed cable net and tensegrity model yielded much lower elastic moduli (10(1)-10(2)Pa) which were consistent with values determined from equilibrium measurements at low applied stress. This suggests that CSK prestress and architecture are the primary determinants of the cell elastic response. The tensegrity model revealed the possibility that buckling of microtubules of the CSK also contributed to cell elasticity.     

Mechanical and failure properties of extracellular matrix sheets as a function of structural protein composition

The goal of this study was to determine how alterations in protein composition of the extracellular matrix (ECM) affect its functional properties. To achieve this, we investigated the changes in the mechanical and failure properties of ECM sheets generated by neonatal rat aortic smooth muscle cells engineered to contain varying amounts of collagen and elastin. Samples underwent static and dynamic mechanical measurements before, during, and after 30 min of elastase digestion followed by a failure test. Microscopic imaging was used to measure thickness at two strain levels to estimate the true stress and moduli in the ECM sheets. We found that adding collagen to the ECM increased the stiffness. However, further increasing collagen content altered matrix organization with a subsequent decrease in the failure strain. We also introduced collagen-related percolation in a nonlinear elastic network model to interpret these results. Additionally, linear elastic moduli correlated with failure stress which may allow the in vivo estimation of the stress tolerance of ECM. We conclude that, in engineered replacement tissues, there is a tradeoff between improved mechanical properties and decreased extensibility, which can impact their effectiveness and how well they match the mechanical properties of native tissue.     

Local rheology of human neutrophils investigated using atomic force microscopy

During the immune response, neutrophils display localized mechanical events by interacting with their environment through the micro-vascular transit, trans-endothelial, and trans-epithelial migration. Nano-mechanical studies of human neutrophils on localized nano-domains could provide the essential information for understanding their immune responsive functions. Using the Atomic Force Microscopy (AFM)-based micro-rheology, we have investigated rheological properties of the adherent human neutrophils on local nano-domains. We have applied the modified Hertz model to obtain the viscoelastic moduli from the relatively thick body regions of the neutrophils. In addition, by using more advanced models to account for the substrate effects, we have successfully characterized the rheological properties of the thin leading and tail regions as well. We found a regional difference in the mechanical compliances of the adherent neutrophils. The central regions of neutrophils were significantly stiffer (1,548 ± 871 Pa) than the regions closer to the leading edge (686 ± 801 Pa), while the leading edge and the tail (494 ± 537 Pa) regions were mechanically indistinguishable. The frequency-dependent elastic and viscous moduli also display a similar regional difference. Over the studied frequency range (100 to 300 Hz), the complex viscoelastic moduli display the partial rubber plateau behavior where the elastic moduli are greater than the viscous moduli for a given frequency. The non-disparaging viscous modulus indicates that the neutrophils display a viscoelastic dynamic behavior rather than a perfect elastic behavior like polymer gels. In addition, we found no regional difference in the structural damping coefficient between the leading edge and the cell body. Thus, we conclude that despite the lower loss and storage moduli, the leading edges of the human neutrophils display partially elastic properties similar to the cell body. These results suggest that the lower elastic moduli in the leading edges are more favorable for the elastic fluctuation of actin filaments, which supports the polymerization of the actin filaments leading to the active protrusion during the immune response.     

Judging Diatoms by Their Cover: Variability in Local Elasticity of Lithodesmium undulatum Undergoing Cell Division

Unique features of diatoms are their intricate cell covers (frustules) made out of hydrated, amorphous silica. The frustule defines and maintains cell shape and protects cells against grazers and pathogens, yet it must allow for cell expansion during growth and division. Other siliceous structures have also evolved in some chain-forming species as means for holding neighboring cells together. Characterization and quantification of mechanical properties of these structures are crucial for the understanding of the relationship between form and function in diatoms, but thus far only a handful of studies have addressed this issue. We conducted micro-indentation experiments, using atomic force microscopy (AFM), to examine local variations in elastic (Young''s) moduli of cells and linking structures in the marine, chain-forming diatom Lithodesmium undulatum. Using a fluorescent tracer that is incorporated into new cell wall components we tested the hypothesis that new siliceous structures differ in elastic modulus from their older counterparts. Results show that the local elastic modulus is a highly dynamic property. Elastic modulus of stained regions was significantly lower than that of unstained regions, suggesting that newly formed cell wall components are generally softer than the ones inherited from the parent cells. This study provides the first evidence of differentiation in local elastic properties in the course of the cell cycle. Hardening of newly formed regions may involve incorporation of additional, possibly organic, material but further studies are needed to elucidate the processes that regulate mechanical properties of the frustule during the cell cycle.     

Effect of Age and Cytoskeletal Elements on the Indentation-Dependent Mechanical Properties of Chondrocytes

Articular cartilage chondrocytes are responsible for the synthesis, maintenance, and turnover of the extracellular matrix, metabolic processes that contribute to the mechanical properties of these cells. Here, we systematically evaluated the effect of age and cytoskeletal disruptors on the mechanical properties of chondrocytes as a function of deformation. We quantified the indentation-dependent mechanical properties of chondrocytes isolated from neonatal (1-day), adult (5-year) and geriatric (12-year) bovine knees using atomic force microscopy (AFM). We also measured the contribution of the actin and intermediate filaments to the indentation-dependent mechanical properties of chondrocytes. By integrating AFM with confocal fluorescent microscopy, we monitored cytoskeletal and biomechanical deformation in transgenic cells (GFP-vimentin and mCherry-actin) under compression. We found that the elastic modulus of chondrocytes in all age groups decreased with increased indentation (15–2000 nm). The elastic modulus of adult chondrocytes was significantly greater than neonatal cells at indentations greater than 500 nm. Viscoelastic moduli (instantaneous and equilibrium) were comparable in all age groups examined; however, the intrinsic viscosity was lower in geriatric chondrocytes than neonatal. Disrupting the actin or the intermediate filament structures altered the mechanical properties of chondrocytes by decreasing the elastic modulus and viscoelastic properties, resulting in a dramatic loss of indentation-dependent response with treatment. Actin and vimentin cytoskeletal structures were monitored using confocal fluorescent microscopy in transgenic cells treated with disruptors, and both treatments had a profound disruptive effect on the actin filaments. Here we show that disrupting the structure of intermediate filaments indirectly altered the configuration of the actin cytoskeleton. These findings underscore the importance of the cytoskeletal elements in the overall mechanical response of chondrocytes, indicating that intermediate filament integrity is key to the non-linear elastic properties of chondrocytes. This study improves our understanding of the mechanical properties of articular cartilage at the single cell level.     

Spatial Mapping of the Biomechanical Properties of the Pericellular Matrix of Articular Cartilage Measured In Situ via Atomic Force Microscopy

In articular cartilage, chondrocytes are surrounded by a narrow region called the pericellular matrix (PCM), which is biochemically, structurally, and mechanically distinct from the bulk extracellular matrix (ECM). Although multiple techniques have been used to measure the mechanical properties of the PCM using isolated chondrons (the PCM with enclosed cells), few studies have measured the biomechanical properties of the PCM in situ. The objective of this study was to quantify the in situ mechanical properties of the PCM and ECM of human, porcine, and murine articular cartilage using atomic force microscopy (AFM). Microscale elastic moduli were quantitatively measured for a region of interest using stiffness mapping, or force-volume mapping, via AFM. This technique was first validated by means of elastomeric models (polyacrylamide or polydimethylsiloxane) of a soft inclusion surrounded by a stiff medium. The elastic properties of the PCM were evaluated for regions surrounding cell voids in the middle/deep zone of sectioned articular cartilage samples. ECM elastic properties were evaluated in regions visually devoid of PCM. Stiffness mapping successfully depicted the spatial arrangement of moduli in both model and cartilage surfaces. The modulus of the PCM was significantly lower than that of the ECM in human, porcine, and murine articular cartilage, with a ratio of PCM to ECM properties of ∼0.35 for all species. These findings are consistent with previous studies of mechanically isolated chondrons, and suggest that stiffness mapping via AFM can provide a means of determining microscale inhomogeneities in the mechanical properties of articular cartilage in situ.     

Endothelial, cardiac muscle and skeletal muscle exhibit different viscous and elastic properties as determined by atomic force microscopy

This study evaluated the hypothesis that, due to functional and structural differences, the apparent elastic modulus and viscous behavior of cardiac and skeletal muscle and vascular endothelium would differ. To accurately determine the elastic modulus, the contribution of probe velocity, indentation depth, and the assumed shape of the probe were examined. Hysteresis was observed at high indentation velocities arising from viscous effects. Irreversible deformation was not observed for endothelial cells and hysteresis was negligible below 1 μm/s. For skeletal muscle and cardiac muscle cells, hysteresis was negligible below 0.25 μm/s. Viscous dissipation for endothelial and cardiac muscle cells was higher than for skeletal muscle cells. The calculated elastic modulus was most sensitive to the assumed probe geometry for the first 60 nm of indentation for the three cell types. Modeling the probe as a blunt cone–spherical cap resulted in variation in elastic modulus with indentation depth that was less than that calculated by treating the probe as a conical tip. Substrate contributions were negligible since the elastic modulus reached a steady value for indentations above 60 nm and the probe never indented more than 10% of the cell thickness. Cardiac cells were the stiffest (100.3±10.7 kPa), the skeletal muscle cells were intermediate (24.7±3.5 kPa), and the endothelial cells were the softest with a range of elastic moduli (1.4±0.1 to 6.8±0.4 kPa) depending on the location of the cell surface tested. Cardiac and skeletal muscle exhibited nonlinear elastic behavior. These passive mechanical properties are generally consistent with the function of these different cell types.     

Measurements of elastic moduli of silicone gel substrates with a microfluidic device

Thin layers of gels with mechanical properties mimicking animal tissues are widely used to study the rigidity sensing of adherent animal cells and to measure forces applied by cells to their substrate with traction force microscopy. The gels are usually based on polyacrylamide and their elastic modulus is measured with an atomic force microscope (AFM). Here we present a simple microfluidic device that generates high shear stresses in a laminar flow above a gel-coated substrate and apply the device to gels with elastic moduli in a range from 0.4 to 300 kPa that are all prepared by mixing two components of a transparent commercial silicone Sylgard 184. The elastic modulus is measured by tracking beads on the gel surface under a wide-field fluorescence microscope without any other specialized equipment. The measurements have small and simple to estimate errors and their results are confirmed by conventional tensile tests. A master curve is obtained relating the mixing ratios of the two components of Sylgard 184 with the resulting elastic moduli of the gels. The rigidity of the silicone gels is less susceptible to effects from drying, swelling, and aging than polyacrylamide gels and can be easily coated with fluorescent tracer particles and with molecules promoting cellular adhesion. This work can lead to broader use of silicone gels in the cell biology laboratory and to improved repeatability and accuracy of cell traction force microscopy and rigidity sensing experiments.     

Vimentin Enhances Cell Elastic Behavior and Protects against Compressive Stress

Vimentin intermediate filament expression is a hallmark of epithelial-to-mesenchymal transitions, and vimentin is involved in the maintenance of cell mechanical properties, cell motility, adhesion, and other signaling pathways. A common feature of vimentin-expressing cells is their routine exposure to mechanical stress. Intermediate filaments are unique among cytoskeletal polymers in resisting large deformations in vitro, yet vimentin’s mechanical role in the cell is not clearly understood. We use atomic force microscopy to compare the viscoelastic properties of normal and vimentin-null (vim^−/−) mouse embryo fibroblasts (mEFs) on substrates of different stiffnesses, spread to different areas, and subjected to different compression patterns. In minimally perturbed mEF, vimentin contributes little to the elastic modulus at any indentation depth in cells spread to average areas. On a hard substrate however, the elastic moduli of maximally spread mEFs are greater than those of vim^−/−mEF. Comparison of the plastic deformation resulting from controlled compression of the cell cortex shows that vimentin’s enhancement of elastic behavior increases with substrate stiffness. The elastic moduli of normal mEFs are more stable over time than those of vim^−/−mEFs when cells are subject to ongoing oscillatory compression, particularly on a soft substrate. In contrast, increasing compressive strain over time shows a greater role for vimentin on a hard substrate. Under both conditions, vim^−/−mEFs exhibit more variable responses, indicating a loss of regulation. Finally, normal mEFs are more contractile in three-dimensional collagen gels when seeded at low density, when cell-matrix contacts dominate, whereas contractility of vim^−/−mEF is greater at higher densities when cell-cell contacts are abundant. Addition of fibronectin to gel constructs equalizes the contractility of the two cell types. These results show that the Young’s moduli of normal and vim^−/−mEFs are substrate stiffness dependent even when the spread area is similar, and that vimentin protects against compressive stress and preserves mechanical integrity by enhancing cell elastic behavior.     

Thermal fluctuations of red blood cell membrane via a constant-area particle-dynamics model

We describe a model of the mechanical properties of the cell plasma membrane using a finite-temperature particle-dynamics simulation of the whole cell, in which a two-dimensional network of virtual particles embedded in a three-dimensional closed surface represents the membrane. The particles interact via harmonic potential and dihedral angle potential and are subject to a constant area constraint. The evolution of the positions of the particles yields the equilibrium state of the membrane and allows determination of the membrane thermal fluctuations and the elastic moduli. We show that time-averaging of the cell-model configurations allows quantitative comparison with experimental data on membrane fluctuations and elastic moduli of the red blood cell.     

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

A numerical study to determine pericellular matrix modulus and evaluate its effects on the micromechanical environment of chondrocytes

Chondrocyte biosynthesis is highly sensitive to mechanical strain. A thin pericellular matrix (PCM) surrounds the cell and plays an important role in mechanotransduction. PCM material properties are difficult to measure directly because of its size and connectivity to both cell and extracellular matrix (ECM). The purpose of this study was to develop a method of calculating linear elastic properties of the PCM using an inverse finite element approach with experimental properties of cell and chondron taken from the literature. Finite element models were constructed of both the equivalent chondron case and the chondrocyte-PCM structure, and a Fibonacci search obtained PCM moduli that matched the ECM strain field between the two cases. The most important result was that ECM strain adjacent to a chondron inclusion was sensitive to the chondron properties and may be used to calculate PCM mechanical properties, consistent with our strain field hypotheses. PCM moduli obtained through this method range from 43 to 240 kPa, were significantly higher than previously published but resulted in only a 0.5-21% decrease in relative effective cell strain. Similarities between effective strain ratios led to the conclusion that matching experimental techniques used to measure cell and PCM properties was more important than absolute values of the properties.